The local $C^1$-density of stable ergodicity

TL;DR

This paper proves that stable ergodicity is densely achievable in the space of conservative partially hyperbolic systems with a robustly non-hyperbolic center, highlighting the prevalence of stable ergodic behavior.

Contribution

It establishes the $C^1$-density of stable ergodicity in a specific class of conservative partially hyperbolic systems with non-hyperbolic centers.

Findings

Stable ergodicity is $C^1$-dense among systems with robust non-hyperbolic centers.

The result applies to conservative partially hyperbolic diffeomorphisms.

It advances understanding of ergodic properties in non-hyperbolic settings.

Abstract

The center bundle of a conservative partially hyperbolic diffeomorphism $f$ is called robustly non-hyperbolic if any conservative diffeomorphism which is $C^1$-close to $f$ has non-hyperbolic center bundle. In this paper, we prove that stable ergodicity is $C^1$-dense among conservative partially hyperbolic systems with robust non-hyperbolic center.